: Mathematics: quadratic irrational

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quadratic irrational

In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as
${\\displaystyle {a+b{\\sqrt {c}} \\over d},}$
for integers $a, b, c, d$ ; with $b$ , $c$ and $d$ non-zero, and with $c$ square-free. When $c$ is positive, we get real quadratic irrational numbers, while a negative $c$ gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Quadratic_irrational_number)

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